# Properties

 Label 1881.2.a.p.1.6 Level $1881$ Weight $2$ Character 1881.1 Self dual yes Analytic conductor $15.020$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1881,2,Mod(1,1881)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1881, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1881.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1881 = 3^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1881.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$15.0198606202$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30$$ x^7 - x^6 - 14*x^5 + 10*x^4 + 59*x^3 - 27*x^2 - 66*x + 30 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 209) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$2.61330$$ of defining polynomial Character $$\chi$$ $$=$$ 1881.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.61330 q^{2} +4.82936 q^{4} -4.07680 q^{5} +3.61829 q^{7} +7.39397 q^{8} +O(q^{10})$$ $$q+2.61330 q^{2} +4.82936 q^{4} -4.07680 q^{5} +3.61829 q^{7} +7.39397 q^{8} -10.6539 q^{10} +1.00000 q^{11} -1.47857 q^{13} +9.45570 q^{14} +9.66398 q^{16} +3.27003 q^{17} +1.00000 q^{19} -19.6883 q^{20} +2.61330 q^{22} +7.45793 q^{23} +11.6203 q^{25} -3.86395 q^{26} +17.4740 q^{28} -1.02535 q^{29} +1.64921 q^{31} +10.4670 q^{32} +8.54558 q^{34} -14.7511 q^{35} -6.71293 q^{37} +2.61330 q^{38} -30.1438 q^{40} +3.92451 q^{41} +5.38113 q^{43} +4.82936 q^{44} +19.4898 q^{46} +3.71597 q^{47} +6.09205 q^{49} +30.3674 q^{50} -7.14054 q^{52} +0.102902 q^{53} -4.07680 q^{55} +26.7536 q^{56} -2.67955 q^{58} -13.2986 q^{59} -6.49664 q^{61} +4.30989 q^{62} +8.02543 q^{64} +6.02783 q^{65} -3.70989 q^{67} +15.7921 q^{68} -38.5490 q^{70} -6.32968 q^{71} -1.37759 q^{73} -17.5429 q^{74} +4.82936 q^{76} +3.61829 q^{77} +13.6725 q^{79} -39.3981 q^{80} +10.2559 q^{82} -5.44061 q^{83} -13.3313 q^{85} +14.0625 q^{86} +7.39397 q^{88} -12.1357 q^{89} -5.34990 q^{91} +36.0170 q^{92} +9.71096 q^{94} -4.07680 q^{95} -13.7910 q^{97} +15.9204 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + q^{2} + 15 q^{4} - 2 q^{5} + 10 q^{7} + 9 q^{8}+O(q^{10})$$ 7 * q + q^2 + 15 * q^4 - 2 * q^5 + 10 * q^7 + 9 * q^8 $$7 q + q^{2} + 15 q^{4} - 2 q^{5} + 10 q^{7} + 9 q^{8} - 6 q^{10} + 7 q^{11} - 4 q^{13} - 6 q^{14} + 27 q^{16} - 2 q^{17} + 7 q^{19} + 4 q^{20} + q^{22} - 10 q^{23} + 9 q^{25} + 8 q^{26} + 26 q^{28} + 18 q^{29} + 24 q^{31} + 49 q^{32} - 6 q^{34} - 8 q^{35} + q^{38} - 2 q^{40} + 12 q^{41} + 2 q^{43} + 15 q^{44} - 4 q^{46} - 8 q^{47} + 17 q^{49} + 33 q^{50} - 60 q^{52} - 2 q^{53} - 2 q^{55} - 26 q^{56} - 8 q^{58} + 10 q^{59} + 14 q^{61} - 14 q^{62} + 55 q^{64} + 14 q^{65} + 8 q^{67} + 18 q^{68} - 66 q^{70} - 10 q^{71} - 6 q^{73} - 26 q^{74} + 15 q^{76} + 10 q^{77} + 52 q^{79} + 12 q^{80} + 24 q^{82} + 10 q^{83} - 12 q^{85} - 8 q^{86} + 9 q^{88} + 12 q^{91} + 24 q^{94} - 2 q^{95} - 24 q^{97} - 19 q^{98}+O(q^{100})$$ 7 * q + q^2 + 15 * q^4 - 2 * q^5 + 10 * q^7 + 9 * q^8 - 6 * q^10 + 7 * q^11 - 4 * q^13 - 6 * q^14 + 27 * q^16 - 2 * q^17 + 7 * q^19 + 4 * q^20 + q^22 - 10 * q^23 + 9 * q^25 + 8 * q^26 + 26 * q^28 + 18 * q^29 + 24 * q^31 + 49 * q^32 - 6 * q^34 - 8 * q^35 + q^38 - 2 * q^40 + 12 * q^41 + 2 * q^43 + 15 * q^44 - 4 * q^46 - 8 * q^47 + 17 * q^49 + 33 * q^50 - 60 * q^52 - 2 * q^53 - 2 * q^55 - 26 * q^56 - 8 * q^58 + 10 * q^59 + 14 * q^61 - 14 * q^62 + 55 * q^64 + 14 * q^65 + 8 * q^67 + 18 * q^68 - 66 * q^70 - 10 * q^71 - 6 * q^73 - 26 * q^74 + 15 * q^76 + 10 * q^77 + 52 * q^79 + 12 * q^80 + 24 * q^82 + 10 * q^83 - 12 * q^85 - 8 * q^86 + 9 * q^88 + 12 * q^91 + 24 * q^94 - 2 * q^95 - 24 * q^97 - 19 * q^98

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.61330 1.84789 0.923943 0.382531i $$-0.124948\pi$$
0.923943 + 0.382531i $$0.124948\pi$$
$$3$$ 0 0
$$4$$ 4.82936 2.41468
$$5$$ −4.07680 −1.82320 −0.911600 0.411078i $$-0.865153\pi$$
−0.911600 + 0.411078i $$0.865153\pi$$
$$6$$ 0 0
$$7$$ 3.61829 1.36759 0.683793 0.729676i $$-0.260329\pi$$
0.683793 + 0.729676i $$0.260329\pi$$
$$8$$ 7.39397 2.61416
$$9$$ 0 0
$$10$$ −10.6539 −3.36907
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −1.47857 −0.410081 −0.205041 0.978753i $$-0.565733\pi$$
−0.205041 + 0.978753i $$0.565733\pi$$
$$14$$ 9.45570 2.52714
$$15$$ 0 0
$$16$$ 9.66398 2.41600
$$17$$ 3.27003 0.793099 0.396549 0.918013i $$-0.370208\pi$$
0.396549 + 0.918013i $$0.370208\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ −19.6883 −4.40244
$$21$$ 0 0
$$22$$ 2.61330 0.557158
$$23$$ 7.45793 1.55509 0.777543 0.628830i $$-0.216466\pi$$
0.777543 + 0.628830i $$0.216466\pi$$
$$24$$ 0 0
$$25$$ 11.6203 2.32406
$$26$$ −3.86395 −0.757783
$$27$$ 0 0
$$28$$ 17.4740 3.30228
$$29$$ −1.02535 −0.190403 −0.0952013 0.995458i $$-0.530349\pi$$
−0.0952013 + 0.995458i $$0.530349\pi$$
$$30$$ 0 0
$$31$$ 1.64921 0.296207 0.148104 0.988972i $$-0.452683\pi$$
0.148104 + 0.988972i $$0.452683\pi$$
$$32$$ 10.4670 1.85032
$$33$$ 0 0
$$34$$ 8.54558 1.46556
$$35$$ −14.7511 −2.49338
$$36$$ 0 0
$$37$$ −6.71293 −1.10360 −0.551799 0.833977i $$-0.686059\pi$$
−0.551799 + 0.833977i $$0.686059\pi$$
$$38$$ 2.61330 0.423934
$$39$$ 0 0
$$40$$ −30.1438 −4.76615
$$41$$ 3.92451 0.612905 0.306453 0.951886i $$-0.400858\pi$$
0.306453 + 0.951886i $$0.400858\pi$$
$$42$$ 0 0
$$43$$ 5.38113 0.820614 0.410307 0.911947i $$-0.365422\pi$$
0.410307 + 0.911947i $$0.365422\pi$$
$$44$$ 4.82936 0.728053
$$45$$ 0 0
$$46$$ 19.4898 2.87362
$$47$$ 3.71597 0.542030 0.271015 0.962575i $$-0.412641\pi$$
0.271015 + 0.962575i $$0.412641\pi$$
$$48$$ 0 0
$$49$$ 6.09205 0.870292
$$50$$ 30.3674 4.29460
$$51$$ 0 0
$$52$$ −7.14054 −0.990215
$$53$$ 0.102902 0.0141347 0.00706733 0.999975i $$-0.497750\pi$$
0.00706733 + 0.999975i $$0.497750\pi$$
$$54$$ 0 0
$$55$$ −4.07680 −0.549716
$$56$$ 26.7536 3.57509
$$57$$ 0 0
$$58$$ −2.67955 −0.351842
$$59$$ −13.2986 −1.73134 −0.865668 0.500619i $$-0.833106\pi$$
−0.865668 + 0.500619i $$0.833106\pi$$
$$60$$ 0 0
$$61$$ −6.49664 −0.831809 −0.415905 0.909408i $$-0.636535\pi$$
−0.415905 + 0.909408i $$0.636535\pi$$
$$62$$ 4.30989 0.547357
$$63$$ 0 0
$$64$$ 8.02543 1.00318
$$65$$ 6.02783 0.747661
$$66$$ 0 0
$$67$$ −3.70989 −0.453235 −0.226618 0.973984i $$-0.572767\pi$$
−0.226618 + 0.973984i $$0.572767\pi$$
$$68$$ 15.7921 1.91508
$$69$$ 0 0
$$70$$ −38.5490 −4.60749
$$71$$ −6.32968 −0.751194 −0.375597 0.926783i $$-0.622562\pi$$
−0.375597 + 0.926783i $$0.622562\pi$$
$$72$$ 0 0
$$73$$ −1.37759 −0.161235 −0.0806173 0.996745i $$-0.525689\pi$$
−0.0806173 + 0.996745i $$0.525689\pi$$
$$74$$ −17.5429 −2.03932
$$75$$ 0 0
$$76$$ 4.82936 0.553965
$$77$$ 3.61829 0.412343
$$78$$ 0 0
$$79$$ 13.6725 1.53828 0.769141 0.639079i $$-0.220684\pi$$
0.769141 + 0.639079i $$0.220684\pi$$
$$80$$ −39.3981 −4.40484
$$81$$ 0 0
$$82$$ 10.2559 1.13258
$$83$$ −5.44061 −0.597184 −0.298592 0.954381i $$-0.596517\pi$$
−0.298592 + 0.954381i $$0.596517\pi$$
$$84$$ 0 0
$$85$$ −13.3313 −1.44598
$$86$$ 14.0625 1.51640
$$87$$ 0 0
$$88$$ 7.39397 0.788200
$$89$$ −12.1357 −1.28638 −0.643191 0.765706i $$-0.722390\pi$$
−0.643191 + 0.765706i $$0.722390\pi$$
$$90$$ 0 0
$$91$$ −5.34990 −0.560822
$$92$$ 36.0170 3.75503
$$93$$ 0 0
$$94$$ 9.71096 1.00161
$$95$$ −4.07680 −0.418271
$$96$$ 0 0
$$97$$ −13.7910 −1.40026 −0.700131 0.714014i $$-0.746875\pi$$
−0.700131 + 0.714014i $$0.746875\pi$$
$$98$$ 15.9204 1.60820
$$99$$ 0 0
$$100$$ 56.1186 5.61186
$$101$$ 11.0029 1.09483 0.547413 0.836863i $$-0.315613\pi$$
0.547413 + 0.836863i $$0.315613\pi$$
$$102$$ 0 0
$$103$$ −4.99191 −0.491867 −0.245934 0.969287i $$-0.579094\pi$$
−0.245934 + 0.969287i $$0.579094\pi$$
$$104$$ −10.9325 −1.07202
$$105$$ 0 0
$$106$$ 0.268914 0.0261192
$$107$$ −7.31345 −0.707018 −0.353509 0.935431i $$-0.615012\pi$$
−0.353509 + 0.935431i $$0.615012\pi$$
$$108$$ 0 0
$$109$$ −1.44482 −0.138389 −0.0691944 0.997603i $$-0.522043\pi$$
−0.0691944 + 0.997603i $$0.522043\pi$$
$$110$$ −10.6539 −1.01581
$$111$$ 0 0
$$112$$ 34.9671 3.30408
$$113$$ 12.0369 1.13234 0.566169 0.824289i $$-0.308425\pi$$
0.566169 + 0.824289i $$0.308425\pi$$
$$114$$ 0 0
$$115$$ −30.4045 −2.83523
$$116$$ −4.95178 −0.459761
$$117$$ 0 0
$$118$$ −34.7534 −3.19931
$$119$$ 11.8319 1.08463
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −16.9777 −1.53709
$$123$$ 0 0
$$124$$ 7.96463 0.715245
$$125$$ −26.9897 −2.41403
$$126$$ 0 0
$$127$$ −4.69692 −0.416784 −0.208392 0.978045i $$-0.566823\pi$$
−0.208392 + 0.978045i $$0.566823\pi$$
$$128$$ 0.0389415 0.00344197
$$129$$ 0 0
$$130$$ 15.7526 1.38159
$$131$$ −3.74466 −0.327173 −0.163586 0.986529i $$-0.552306\pi$$
−0.163586 + 0.986529i $$0.552306\pi$$
$$132$$ 0 0
$$133$$ 3.61829 0.313746
$$134$$ −9.69507 −0.837526
$$135$$ 0 0
$$136$$ 24.1785 2.07329
$$137$$ 15.7595 1.34643 0.673213 0.739449i $$-0.264914\pi$$
0.673213 + 0.739449i $$0.264914\pi$$
$$138$$ 0 0
$$139$$ −2.52822 −0.214440 −0.107220 0.994235i $$-0.534195\pi$$
−0.107220 + 0.994235i $$0.534195\pi$$
$$140$$ −71.2381 −6.02072
$$141$$ 0 0
$$142$$ −16.5414 −1.38812
$$143$$ −1.47857 −0.123644
$$144$$ 0 0
$$145$$ 4.18015 0.347142
$$146$$ −3.60006 −0.297943
$$147$$ 0 0
$$148$$ −32.4191 −2.66484
$$149$$ −1.84902 −0.151477 −0.0757387 0.997128i $$-0.524131\pi$$
−0.0757387 + 0.997128i $$0.524131\pi$$
$$150$$ 0 0
$$151$$ −15.3184 −1.24659 −0.623296 0.781986i $$-0.714207\pi$$
−0.623296 + 0.781986i $$0.714207\pi$$
$$152$$ 7.39397 0.599730
$$153$$ 0 0
$$154$$ 9.45570 0.761962
$$155$$ −6.72351 −0.540045
$$156$$ 0 0
$$157$$ 24.6631 1.96833 0.984165 0.177254i $$-0.0567215\pi$$
0.984165 + 0.177254i $$0.0567215\pi$$
$$158$$ 35.7305 2.84257
$$159$$ 0 0
$$160$$ −42.6718 −3.37350
$$161$$ 26.9850 2.12671
$$162$$ 0 0
$$163$$ −3.72149 −0.291490 −0.145745 0.989322i $$-0.546558\pi$$
−0.145745 + 0.989322i $$0.546558\pi$$
$$164$$ 18.9529 1.47997
$$165$$ 0 0
$$166$$ −14.2180 −1.10353
$$167$$ −2.64758 −0.204876 −0.102438 0.994739i $$-0.532664\pi$$
−0.102438 + 0.994739i $$0.532664\pi$$
$$168$$ 0 0
$$169$$ −10.8138 −0.831833
$$170$$ −34.8386 −2.67200
$$171$$ 0 0
$$172$$ 25.9874 1.98152
$$173$$ −7.34552 −0.558469 −0.279235 0.960223i $$-0.590081\pi$$
−0.279235 + 0.960223i $$0.590081\pi$$
$$174$$ 0 0
$$175$$ 42.0457 3.17835
$$176$$ 9.66398 0.728450
$$177$$ 0 0
$$178$$ −31.7143 −2.37708
$$179$$ −9.55394 −0.714095 −0.357047 0.934086i $$-0.616217\pi$$
−0.357047 + 0.934086i $$0.616217\pi$$
$$180$$ 0 0
$$181$$ 6.02638 0.447937 0.223969 0.974596i $$-0.428099\pi$$
0.223969 + 0.974596i $$0.428099\pi$$
$$182$$ −13.9809 −1.03633
$$183$$ 0 0
$$184$$ 55.1437 4.06525
$$185$$ 27.3673 2.01208
$$186$$ 0 0
$$187$$ 3.27003 0.239128
$$188$$ 17.9458 1.30883
$$189$$ 0 0
$$190$$ −10.6539 −0.772917
$$191$$ −17.7069 −1.28123 −0.640613 0.767864i $$-0.721320\pi$$
−0.640613 + 0.767864i $$0.721320\pi$$
$$192$$ 0 0
$$193$$ 3.69348 0.265863 0.132931 0.991125i $$-0.457561\pi$$
0.132931 + 0.991125i $$0.457561\pi$$
$$194$$ −36.0400 −2.58752
$$195$$ 0 0
$$196$$ 29.4207 2.10148
$$197$$ 25.7789 1.83667 0.918336 0.395802i $$-0.129533\pi$$
0.918336 + 0.395802i $$0.129533\pi$$
$$198$$ 0 0
$$199$$ 18.8953 1.33945 0.669726 0.742608i $$-0.266411\pi$$
0.669726 + 0.742608i $$0.266411\pi$$
$$200$$ 85.9202 6.07548
$$201$$ 0 0
$$202$$ 28.7538 2.02311
$$203$$ −3.71002 −0.260392
$$204$$ 0 0
$$205$$ −15.9994 −1.11745
$$206$$ −13.0454 −0.908914
$$207$$ 0 0
$$208$$ −14.2889 −0.990755
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ −10.1993 −0.702150 −0.351075 0.936347i $$-0.614184\pi$$
−0.351075 + 0.936347i $$0.614184\pi$$
$$212$$ 0.496950 0.0341306
$$213$$ 0 0
$$214$$ −19.1123 −1.30649
$$215$$ −21.9378 −1.49614
$$216$$ 0 0
$$217$$ 5.96733 0.405089
$$218$$ −3.77576 −0.255727
$$219$$ 0 0
$$220$$ −19.6883 −1.32739
$$221$$ −4.83497 −0.325235
$$222$$ 0 0
$$223$$ −0.262700 −0.0175917 −0.00879584 0.999961i $$-0.502800\pi$$
−0.00879584 + 0.999961i $$0.502800\pi$$
$$224$$ 37.8726 2.53047
$$225$$ 0 0
$$226$$ 31.4561 2.09243
$$227$$ −27.3256 −1.81367 −0.906834 0.421489i $$-0.861508\pi$$
−0.906834 + 0.421489i $$0.861508\pi$$
$$228$$ 0 0
$$229$$ −5.53171 −0.365546 −0.182773 0.983155i $$-0.558507\pi$$
−0.182773 + 0.983155i $$0.558507\pi$$
$$230$$ −79.4562 −5.23918
$$231$$ 0 0
$$232$$ −7.58141 −0.497744
$$233$$ −27.4733 −1.79984 −0.899918 0.436059i $$-0.856374\pi$$
−0.899918 + 0.436059i $$0.856374\pi$$
$$234$$ 0 0
$$235$$ −15.1493 −0.988229
$$236$$ −64.2239 −4.18062
$$237$$ 0 0
$$238$$ 30.9204 2.00427
$$239$$ −1.40339 −0.0907781 −0.0453890 0.998969i $$-0.514453\pi$$
−0.0453890 + 0.998969i $$0.514453\pi$$
$$240$$ 0 0
$$241$$ −20.7696 −1.33789 −0.668944 0.743312i $$-0.733254\pi$$
−0.668944 + 0.743312i $$0.733254\pi$$
$$242$$ 2.61330 0.167990
$$243$$ 0 0
$$244$$ −31.3746 −2.00855
$$245$$ −24.8361 −1.58672
$$246$$ 0 0
$$247$$ −1.47857 −0.0940791
$$248$$ 12.1942 0.774334
$$249$$ 0 0
$$250$$ −70.5322 −4.46085
$$251$$ 1.29936 0.0820148 0.0410074 0.999159i $$-0.486943\pi$$
0.0410074 + 0.999159i $$0.486943\pi$$
$$252$$ 0 0
$$253$$ 7.45793 0.468876
$$254$$ −12.2745 −0.770169
$$255$$ 0 0
$$256$$ −15.9491 −0.996819
$$257$$ −3.41219 −0.212847 −0.106423 0.994321i $$-0.533940\pi$$
−0.106423 + 0.994321i $$0.533940\pi$$
$$258$$ 0 0
$$259$$ −24.2893 −1.50927
$$260$$ 29.1106 1.80536
$$261$$ 0 0
$$262$$ −9.78594 −0.604577
$$263$$ −14.2418 −0.878189 −0.439094 0.898441i $$-0.644701\pi$$
−0.439094 + 0.898441i $$0.644701\pi$$
$$264$$ 0 0
$$265$$ −0.419510 −0.0257703
$$266$$ 9.45570 0.579766
$$267$$ 0 0
$$268$$ −17.9164 −1.09442
$$269$$ −5.39477 −0.328925 −0.164462 0.986383i $$-0.552589\pi$$
−0.164462 + 0.986383i $$0.552589\pi$$
$$270$$ 0 0
$$271$$ −11.0624 −0.671995 −0.335998 0.941863i $$-0.609073\pi$$
−0.335998 + 0.941863i $$0.609073\pi$$
$$272$$ 31.6015 1.91612
$$273$$ 0 0
$$274$$ 41.1844 2.48804
$$275$$ 11.6203 0.700731
$$276$$ 0 0
$$277$$ 14.0808 0.846036 0.423018 0.906121i $$-0.360971\pi$$
0.423018 + 0.906121i $$0.360971\pi$$
$$278$$ −6.60700 −0.396261
$$279$$ 0 0
$$280$$ −109.069 −6.51812
$$281$$ 10.8199 0.645459 0.322729 0.946491i $$-0.395400\pi$$
0.322729 + 0.946491i $$0.395400\pi$$
$$282$$ 0 0
$$283$$ 1.90947 0.113506 0.0567532 0.998388i $$-0.481925\pi$$
0.0567532 + 0.998388i $$0.481925\pi$$
$$284$$ −30.5683 −1.81389
$$285$$ 0 0
$$286$$ −3.86395 −0.228480
$$287$$ 14.2000 0.838201
$$288$$ 0 0
$$289$$ −6.30690 −0.370994
$$290$$ 10.9240 0.641479
$$291$$ 0 0
$$292$$ −6.65287 −0.389330
$$293$$ −3.63550 −0.212388 −0.106194 0.994345i $$-0.533867\pi$$
−0.106194 + 0.994345i $$0.533867\pi$$
$$294$$ 0 0
$$295$$ 54.2159 3.15657
$$296$$ −49.6352 −2.88499
$$297$$ 0 0
$$298$$ −4.83205 −0.279913
$$299$$ −11.0271 −0.637712
$$300$$ 0 0
$$301$$ 19.4705 1.12226
$$302$$ −40.0316 −2.30356
$$303$$ 0 0
$$304$$ 9.66398 0.554267
$$305$$ 26.4855 1.51656
$$306$$ 0 0
$$307$$ −23.5329 −1.34310 −0.671548 0.740961i $$-0.734370\pi$$
−0.671548 + 0.740961i $$0.734370\pi$$
$$308$$ 17.4740 0.995675
$$309$$ 0 0
$$310$$ −17.5706 −0.997941
$$311$$ 16.0026 0.907425 0.453713 0.891148i $$-0.350099\pi$$
0.453713 + 0.891148i $$0.350099\pi$$
$$312$$ 0 0
$$313$$ 16.0034 0.904566 0.452283 0.891875i $$-0.350610\pi$$
0.452283 + 0.891875i $$0.350610\pi$$
$$314$$ 64.4522 3.63725
$$315$$ 0 0
$$316$$ 66.0296 3.71446
$$317$$ 20.8766 1.17255 0.586273 0.810113i $$-0.300595\pi$$
0.586273 + 0.810113i $$0.300595\pi$$
$$318$$ 0 0
$$319$$ −1.02535 −0.0574086
$$320$$ −32.7181 −1.82900
$$321$$ 0 0
$$322$$ 70.5199 3.92992
$$323$$ 3.27003 0.181949
$$324$$ 0 0
$$325$$ −17.1814 −0.953054
$$326$$ −9.72539 −0.538640
$$327$$ 0 0
$$328$$ 29.0177 1.60224
$$329$$ 13.4455 0.741273
$$330$$ 0 0
$$331$$ 15.1136 0.830721 0.415360 0.909657i $$-0.363655\pi$$
0.415360 + 0.909657i $$0.363655\pi$$
$$332$$ −26.2746 −1.44201
$$333$$ 0 0
$$334$$ −6.91893 −0.378587
$$335$$ 15.1245 0.826339
$$336$$ 0 0
$$337$$ −12.2766 −0.668751 −0.334376 0.942440i $$-0.608525\pi$$
−0.334376 + 0.942440i $$0.608525\pi$$
$$338$$ −28.2598 −1.53713
$$339$$ 0 0
$$340$$ −64.3814 −3.49157
$$341$$ 1.64921 0.0893098
$$342$$ 0 0
$$343$$ −3.28525 −0.177387
$$344$$ 39.7879 2.14522
$$345$$ 0 0
$$346$$ −19.1961 −1.03199
$$347$$ 30.9067 1.65916 0.829580 0.558387i $$-0.188580\pi$$
0.829580 + 0.558387i $$0.188580\pi$$
$$348$$ 0 0
$$349$$ −23.9024 −1.27947 −0.639733 0.768597i $$-0.720955\pi$$
−0.639733 + 0.768597i $$0.720955\pi$$
$$350$$ 109.878 5.87323
$$351$$ 0 0
$$352$$ 10.4670 0.557892
$$353$$ −15.0158 −0.799210 −0.399605 0.916687i $$-0.630853\pi$$
−0.399605 + 0.916687i $$0.630853\pi$$
$$354$$ 0 0
$$355$$ 25.8048 1.36958
$$356$$ −58.6076 −3.10620
$$357$$ 0 0
$$358$$ −24.9673 −1.31957
$$359$$ −14.0826 −0.743251 −0.371626 0.928383i $$-0.621200\pi$$
−0.371626 + 0.928383i $$0.621200\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 15.7488 0.827737
$$363$$ 0 0
$$364$$ −25.8366 −1.35420
$$365$$ 5.61616 0.293963
$$366$$ 0 0
$$367$$ 21.3142 1.11259 0.556296 0.830984i $$-0.312222\pi$$
0.556296 + 0.830984i $$0.312222\pi$$
$$368$$ 72.0733 3.75708
$$369$$ 0 0
$$370$$ 71.5190 3.71810
$$371$$ 0.372329 0.0193304
$$372$$ 0 0
$$373$$ −2.42088 −0.125349 −0.0626743 0.998034i $$-0.519963\pi$$
−0.0626743 + 0.998034i $$0.519963\pi$$
$$374$$ 8.54558 0.441882
$$375$$ 0 0
$$376$$ 27.4758 1.41696
$$377$$ 1.51605 0.0780806
$$378$$ 0 0
$$379$$ 8.87535 0.455896 0.227948 0.973673i $$-0.426798\pi$$
0.227948 + 0.973673i $$0.426798\pi$$
$$380$$ −19.6883 −1.00999
$$381$$ 0 0
$$382$$ −46.2735 −2.36756
$$383$$ 3.54065 0.180919 0.0904595 0.995900i $$-0.471166\pi$$
0.0904595 + 0.995900i $$0.471166\pi$$
$$384$$ 0 0
$$385$$ −14.7511 −0.751784
$$386$$ 9.65219 0.491284
$$387$$ 0 0
$$388$$ −66.6016 −3.38118
$$389$$ 16.7041 0.846933 0.423466 0.905912i $$-0.360813\pi$$
0.423466 + 0.905912i $$0.360813\pi$$
$$390$$ 0 0
$$391$$ 24.3877 1.23334
$$392$$ 45.0444 2.27509
$$393$$ 0 0
$$394$$ 67.3681 3.39396
$$395$$ −55.7403 −2.80460
$$396$$ 0 0
$$397$$ 28.2073 1.41568 0.707841 0.706372i $$-0.249669\pi$$
0.707841 + 0.706372i $$0.249669\pi$$
$$398$$ 49.3792 2.47515
$$399$$ 0 0
$$400$$ 112.298 5.61492
$$401$$ 36.2078 1.80813 0.904065 0.427395i $$-0.140569\pi$$
0.904065 + 0.427395i $$0.140569\pi$$
$$402$$ 0 0
$$403$$ −2.43847 −0.121469
$$404$$ 53.1368 2.64365
$$405$$ 0 0
$$406$$ −9.69540 −0.481175
$$407$$ −6.71293 −0.332748
$$408$$ 0 0
$$409$$ 36.9236 1.82576 0.912878 0.408233i $$-0.133855\pi$$
0.912878 + 0.408233i $$0.133855\pi$$
$$410$$ −41.8114 −2.06492
$$411$$ 0 0
$$412$$ −24.1077 −1.18770
$$413$$ −48.1184 −2.36775
$$414$$ 0 0
$$415$$ 22.1803 1.08879
$$416$$ −15.4762 −0.758781
$$417$$ 0 0
$$418$$ 2.61330 0.127821
$$419$$ −18.0690 −0.882726 −0.441363 0.897329i $$-0.645505\pi$$
−0.441363 + 0.897329i $$0.645505\pi$$
$$420$$ 0 0
$$421$$ −8.57629 −0.417983 −0.208991 0.977918i $$-0.567018\pi$$
−0.208991 + 0.977918i $$0.567018\pi$$
$$422$$ −26.6539 −1.29749
$$423$$ 0 0
$$424$$ 0.760853 0.0369503
$$425$$ 37.9987 1.84321
$$426$$ 0 0
$$427$$ −23.5067 −1.13757
$$428$$ −35.3193 −1.70722
$$429$$ 0 0
$$430$$ −57.3301 −2.76470
$$431$$ −4.28147 −0.206231 −0.103116 0.994669i $$-0.532881\pi$$
−0.103116 + 0.994669i $$0.532881\pi$$
$$432$$ 0 0
$$433$$ 18.2035 0.874804 0.437402 0.899266i $$-0.355899\pi$$
0.437402 + 0.899266i $$0.355899\pi$$
$$434$$ 15.5945 0.748558
$$435$$ 0 0
$$436$$ −6.97756 −0.334165
$$437$$ 7.45793 0.356761
$$438$$ 0 0
$$439$$ 29.4442 1.40529 0.702647 0.711538i $$-0.252001\pi$$
0.702647 + 0.711538i $$0.252001\pi$$
$$440$$ −30.1438 −1.43705
$$441$$ 0 0
$$442$$ −12.6352 −0.600997
$$443$$ 24.3633 1.15753 0.578767 0.815493i $$-0.303534\pi$$
0.578767 + 0.815493i $$0.303534\pi$$
$$444$$ 0 0
$$445$$ 49.4748 2.34533
$$446$$ −0.686514 −0.0325074
$$447$$ 0 0
$$448$$ 29.0384 1.37193
$$449$$ −38.7776 −1.83003 −0.915015 0.403421i $$-0.867821\pi$$
−0.915015 + 0.403421i $$0.867821\pi$$
$$450$$ 0 0
$$451$$ 3.92451 0.184798
$$452$$ 58.1306 2.73423
$$453$$ 0 0
$$454$$ −71.4102 −3.35145
$$455$$ 21.8105 1.02249
$$456$$ 0 0
$$457$$ −7.47672 −0.349746 −0.174873 0.984591i $$-0.555952\pi$$
−0.174873 + 0.984591i $$0.555952\pi$$
$$458$$ −14.4560 −0.675487
$$459$$ 0 0
$$460$$ −146.834 −6.84618
$$461$$ −15.9782 −0.744181 −0.372091 0.928196i $$-0.621359\pi$$
−0.372091 + 0.928196i $$0.621359\pi$$
$$462$$ 0 0
$$463$$ −6.10221 −0.283594 −0.141797 0.989896i $$-0.545288\pi$$
−0.141797 + 0.989896i $$0.545288\pi$$
$$464$$ −9.90896 −0.460012
$$465$$ 0 0
$$466$$ −71.7961 −3.32589
$$467$$ 26.6373 1.23263 0.616313 0.787502i $$-0.288626\pi$$
0.616313 + 0.787502i $$0.288626\pi$$
$$468$$ 0 0
$$469$$ −13.4235 −0.619838
$$470$$ −39.5896 −1.82613
$$471$$ 0 0
$$472$$ −98.3298 −4.52599
$$473$$ 5.38113 0.247424
$$474$$ 0 0
$$475$$ 11.6203 0.533176
$$476$$ 57.1406 2.61904
$$477$$ 0 0
$$478$$ −3.66750 −0.167747
$$479$$ 16.2094 0.740626 0.370313 0.928907i $$-0.379250\pi$$
0.370313 + 0.928907i $$0.379250\pi$$
$$480$$ 0 0
$$481$$ 9.92553 0.452565
$$482$$ −54.2773 −2.47226
$$483$$ 0 0
$$484$$ 4.82936 0.219516
$$485$$ 56.2231 2.55296
$$486$$ 0 0
$$487$$ −4.82448 −0.218618 −0.109309 0.994008i $$-0.534864\pi$$
−0.109309 + 0.994008i $$0.534864\pi$$
$$488$$ −48.0360 −2.17449
$$489$$ 0 0
$$490$$ −64.9042 −2.93207
$$491$$ 8.53579 0.385215 0.192607 0.981276i $$-0.438306\pi$$
0.192607 + 0.981276i $$0.438306\pi$$
$$492$$ 0 0
$$493$$ −3.35293 −0.151008
$$494$$ −3.86395 −0.173847
$$495$$ 0 0
$$496$$ 15.9380 0.715635
$$497$$ −22.9026 −1.02732
$$498$$ 0 0
$$499$$ −13.4325 −0.601319 −0.300660 0.953732i $$-0.597207\pi$$
−0.300660 + 0.953732i $$0.597207\pi$$
$$500$$ −130.343 −5.82910
$$501$$ 0 0
$$502$$ 3.39562 0.151554
$$503$$ 1.66487 0.0742327 0.0371163 0.999311i $$-0.488183\pi$$
0.0371163 + 0.999311i $$0.488183\pi$$
$$504$$ 0 0
$$505$$ −44.8565 −1.99609
$$506$$ 19.4898 0.866429
$$507$$ 0 0
$$508$$ −22.6831 −1.00640
$$509$$ 14.9684 0.663463 0.331731 0.943374i $$-0.392367\pi$$
0.331731 + 0.943374i $$0.392367\pi$$
$$510$$ 0 0
$$511$$ −4.98452 −0.220502
$$512$$ −41.7577 −1.84545
$$513$$ 0 0
$$514$$ −8.91709 −0.393316
$$515$$ 20.3510 0.896772
$$516$$ 0 0
$$517$$ 3.71597 0.163428
$$518$$ −63.4754 −2.78895
$$519$$ 0 0
$$520$$ 44.5696 1.95451
$$521$$ 7.89123 0.345721 0.172861 0.984946i $$-0.444699\pi$$
0.172861 + 0.984946i $$0.444699\pi$$
$$522$$ 0 0
$$523$$ −22.3062 −0.975381 −0.487690 0.873017i $$-0.662160\pi$$
−0.487690 + 0.873017i $$0.662160\pi$$
$$524$$ −18.0843 −0.790017
$$525$$ 0 0
$$526$$ −37.2182 −1.62279
$$527$$ 5.39297 0.234922
$$528$$ 0 0
$$529$$ 32.6207 1.41829
$$530$$ −1.09631 −0.0476206
$$531$$ 0 0
$$532$$ 17.4740 0.757595
$$533$$ −5.80266 −0.251341
$$534$$ 0 0
$$535$$ 29.8155 1.28904
$$536$$ −27.4308 −1.18483
$$537$$ 0 0
$$538$$ −14.0982 −0.607815
$$539$$ 6.09205 0.262403
$$540$$ 0 0
$$541$$ 38.9694 1.67542 0.837712 0.546112i $$-0.183893\pi$$
0.837712 + 0.546112i $$0.183893\pi$$
$$542$$ −28.9095 −1.24177
$$543$$ 0 0
$$544$$ 34.2273 1.46749
$$545$$ 5.89025 0.252311
$$546$$ 0 0
$$547$$ 37.7503 1.61409 0.807043 0.590492i $$-0.201066\pi$$
0.807043 + 0.590492i $$0.201066\pi$$
$$548$$ 76.1083 3.25119
$$549$$ 0 0
$$550$$ 30.3674 1.29487
$$551$$ −1.02535 −0.0436814
$$552$$ 0 0
$$553$$ 49.4713 2.10373
$$554$$ 36.7975 1.56338
$$555$$ 0 0
$$556$$ −12.2097 −0.517805
$$557$$ 3.17436 0.134502 0.0672511 0.997736i $$-0.478577\pi$$
0.0672511 + 0.997736i $$0.478577\pi$$
$$558$$ 0 0
$$559$$ −7.95637 −0.336519
$$560$$ −142.554 −6.02401
$$561$$ 0 0
$$562$$ 28.2756 1.19273
$$563$$ 19.9431 0.840503 0.420252 0.907408i $$-0.361942\pi$$
0.420252 + 0.907408i $$0.361942\pi$$
$$564$$ 0 0
$$565$$ −49.0721 −2.06448
$$566$$ 4.99004 0.209747
$$567$$ 0 0
$$568$$ −46.8015 −1.96375
$$569$$ −36.6424 −1.53613 −0.768064 0.640374i $$-0.778780\pi$$
−0.768064 + 0.640374i $$0.778780\pi$$
$$570$$ 0 0
$$571$$ 11.5300 0.482515 0.241258 0.970461i $$-0.422440\pi$$
0.241258 + 0.970461i $$0.422440\pi$$
$$572$$ −7.14054 −0.298561
$$573$$ 0 0
$$574$$ 37.1090 1.54890
$$575$$ 86.6634 3.61411
$$576$$ 0 0
$$577$$ 28.5590 1.18893 0.594463 0.804123i $$-0.297365\pi$$
0.594463 + 0.804123i $$0.297365\pi$$
$$578$$ −16.4818 −0.685554
$$579$$ 0 0
$$580$$ 20.1874 0.838237
$$581$$ −19.6857 −0.816701
$$582$$ 0 0
$$583$$ 0.102902 0.00426176
$$584$$ −10.1859 −0.421494
$$585$$ 0 0
$$586$$ −9.50067 −0.392469
$$587$$ −18.1461 −0.748969 −0.374484 0.927233i $$-0.622180\pi$$
−0.374484 + 0.927233i $$0.622180\pi$$
$$588$$ 0 0
$$589$$ 1.64921 0.0679546
$$590$$ 141.683 5.83298
$$591$$ 0 0
$$592$$ −64.8736 −2.66629
$$593$$ −15.3085 −0.628644 −0.314322 0.949316i $$-0.601777\pi$$
−0.314322 + 0.949316i $$0.601777\pi$$
$$594$$ 0 0
$$595$$ −48.2364 −1.97750
$$596$$ −8.92957 −0.365769
$$597$$ 0 0
$$598$$ −28.8171 −1.17842
$$599$$ −17.8806 −0.730580 −0.365290 0.930894i $$-0.619030\pi$$
−0.365290 + 0.930894i $$0.619030\pi$$
$$600$$ 0 0
$$601$$ −32.5803 −1.32898 −0.664489 0.747298i $$-0.731351\pi$$
−0.664489 + 0.747298i $$0.731351\pi$$
$$602$$ 50.8823 2.07381
$$603$$ 0 0
$$604$$ −73.9779 −3.01012
$$605$$ −4.07680 −0.165746
$$606$$ 0 0
$$607$$ 43.6494 1.77167 0.885837 0.463997i $$-0.153585\pi$$
0.885837 + 0.463997i $$0.153585\pi$$
$$608$$ 10.4670 0.424492
$$609$$ 0 0
$$610$$ 69.2147 2.80242
$$611$$ −5.49432 −0.222276
$$612$$ 0 0
$$613$$ 0.843061 0.0340509 0.0170254 0.999855i $$-0.494580\pi$$
0.0170254 + 0.999855i $$0.494580\pi$$
$$614$$ −61.4987 −2.48189
$$615$$ 0 0
$$616$$ 26.7536 1.07793
$$617$$ −4.89882 −0.197219 −0.0986094 0.995126i $$-0.531439\pi$$
−0.0986094 + 0.995126i $$0.531439\pi$$
$$618$$ 0 0
$$619$$ −14.8704 −0.597691 −0.298846 0.954301i $$-0.596602\pi$$
−0.298846 + 0.954301i $$0.596602\pi$$
$$620$$ −32.4702 −1.30404
$$621$$ 0 0
$$622$$ 41.8197 1.67682
$$623$$ −43.9105 −1.75924
$$624$$ 0 0
$$625$$ 51.9299 2.07720
$$626$$ 41.8217 1.67153
$$627$$ 0 0
$$628$$ 119.107 4.75289
$$629$$ −21.9515 −0.875263
$$630$$ 0 0
$$631$$ 2.83922 0.113027 0.0565137 0.998402i $$-0.482002\pi$$
0.0565137 + 0.998402i $$0.482002\pi$$
$$632$$ 101.094 4.02132
$$633$$ 0 0
$$634$$ 54.5569 2.16673
$$635$$ 19.1484 0.759881
$$636$$ 0 0
$$637$$ −9.00751 −0.356891
$$638$$ −2.67955 −0.106084
$$639$$ 0 0
$$640$$ −0.158757 −0.00627541
$$641$$ −20.6746 −0.816599 −0.408299 0.912848i $$-0.633878\pi$$
−0.408299 + 0.912848i $$0.633878\pi$$
$$642$$ 0 0
$$643$$ −24.6254 −0.971130 −0.485565 0.874201i $$-0.661386\pi$$
−0.485565 + 0.874201i $$0.661386\pi$$
$$644$$ 130.320 5.13533
$$645$$ 0 0
$$646$$ 8.54558 0.336222
$$647$$ −45.3626 −1.78339 −0.891693 0.452640i $$-0.850482\pi$$
−0.891693 + 0.452640i $$0.850482\pi$$
$$648$$ 0 0
$$649$$ −13.2986 −0.522017
$$650$$ −44.9003 −1.76113
$$651$$ 0 0
$$652$$ −17.9724 −0.703854
$$653$$ −26.1012 −1.02142 −0.510710 0.859753i $$-0.670617\pi$$
−0.510710 + 0.859753i $$0.670617\pi$$
$$654$$ 0 0
$$655$$ 15.2662 0.596501
$$656$$ 37.9264 1.48078
$$657$$ 0 0
$$658$$ 35.1371 1.36979
$$659$$ 45.8507 1.78609 0.893045 0.449967i $$-0.148564\pi$$
0.893045 + 0.449967i $$0.148564\pi$$
$$660$$ 0 0
$$661$$ −15.4225 −0.599864 −0.299932 0.953961i $$-0.596964\pi$$
−0.299932 + 0.953961i $$0.596964\pi$$
$$662$$ 39.4965 1.53508
$$663$$ 0 0
$$664$$ −40.2277 −1.56114
$$665$$ −14.7511 −0.572022
$$666$$ 0 0
$$667$$ −7.64698 −0.296092
$$668$$ −12.7861 −0.494709
$$669$$ 0 0
$$670$$ 39.5249 1.52698
$$671$$ −6.49664 −0.250800
$$672$$ 0 0
$$673$$ −37.8633 −1.45952 −0.729762 0.683702i $$-0.760369\pi$$
−0.729762 + 0.683702i $$0.760369\pi$$
$$674$$ −32.0826 −1.23578
$$675$$ 0 0
$$676$$ −52.2239 −2.00861
$$677$$ 25.3510 0.974320 0.487160 0.873313i $$-0.338033\pi$$
0.487160 + 0.873313i $$0.338033\pi$$
$$678$$ 0 0
$$679$$ −49.8998 −1.91498
$$680$$ −98.5710 −3.78003
$$681$$ 0 0
$$682$$ 4.30989 0.165034
$$683$$ 21.7513 0.832290 0.416145 0.909298i $$-0.363381\pi$$
0.416145 + 0.909298i $$0.363381\pi$$
$$684$$ 0 0
$$685$$ −64.2484 −2.45480
$$686$$ −8.58534 −0.327790
$$687$$ 0 0
$$688$$ 52.0031 1.98260
$$689$$ −0.152147 −0.00579636
$$690$$ 0 0
$$691$$ 17.3051 0.658318 0.329159 0.944275i $$-0.393235\pi$$
0.329159 + 0.944275i $$0.393235\pi$$
$$692$$ −35.4741 −1.34852
$$693$$ 0 0
$$694$$ 80.7687 3.06594
$$695$$ 10.3070 0.390968
$$696$$ 0 0
$$697$$ 12.8333 0.486095
$$698$$ −62.4643 −2.36431
$$699$$ 0 0
$$700$$ 203.054 7.67470
$$701$$ 29.6923 1.12146 0.560732 0.827997i $$-0.310520\pi$$
0.560732 + 0.827997i $$0.310520\pi$$
$$702$$ 0 0
$$703$$ −6.71293 −0.253183
$$704$$ 8.02543 0.302470
$$705$$ 0 0
$$706$$ −39.2409 −1.47685
$$707$$ 39.8116 1.49727
$$708$$ 0 0
$$709$$ −21.4898 −0.807067 −0.403534 0.914965i $$-0.632218\pi$$
−0.403534 + 0.914965i $$0.632218\pi$$
$$710$$ 67.4359 2.53082
$$711$$ 0 0
$$712$$ −89.7310 −3.36281
$$713$$ 12.2997 0.460627
$$714$$ 0 0
$$715$$ 6.02783 0.225428
$$716$$ −46.1394 −1.72431
$$717$$ 0 0
$$718$$ −36.8021 −1.37344
$$719$$ −9.61388 −0.358537 −0.179269 0.983800i $$-0.557373\pi$$
−0.179269 + 0.983800i $$0.557373\pi$$
$$720$$ 0 0
$$721$$ −18.0622 −0.672671
$$722$$ 2.61330 0.0972571
$$723$$ 0 0
$$724$$ 29.1036 1.08163
$$725$$ −11.9149 −0.442507
$$726$$ 0 0
$$727$$ 6.84046 0.253699 0.126849 0.991922i $$-0.459514\pi$$
0.126849 + 0.991922i $$0.459514\pi$$
$$728$$ −39.5570 −1.46608
$$729$$ 0 0
$$730$$ 14.6767 0.543210
$$731$$ 17.5965 0.650828
$$732$$ 0 0
$$733$$ 38.2277 1.41197 0.705987 0.708225i $$-0.250504\pi$$
0.705987 + 0.708225i $$0.250504\pi$$
$$734$$ 55.7005 2.05594
$$735$$ 0 0
$$736$$ 78.0620 2.87740
$$737$$ −3.70989 −0.136656
$$738$$ 0 0
$$739$$ −17.7157 −0.651682 −0.325841 0.945425i $$-0.605647\pi$$
−0.325841 + 0.945425i $$0.605647\pi$$
$$740$$ 132.166 4.85853
$$741$$ 0 0
$$742$$ 0.973009 0.0357203
$$743$$ −21.3028 −0.781525 −0.390763 0.920491i $$-0.627789\pi$$
−0.390763 + 0.920491i $$0.627789\pi$$
$$744$$ 0 0
$$745$$ 7.53808 0.276174
$$746$$ −6.32650 −0.231630
$$747$$ 0 0
$$748$$ 15.7921 0.577418
$$749$$ −26.4622 −0.966908
$$750$$ 0 0
$$751$$ 25.1552 0.917926 0.458963 0.888455i $$-0.348221\pi$$
0.458963 + 0.888455i $$0.348221\pi$$
$$752$$ 35.9111 1.30954
$$753$$ 0 0
$$754$$ 3.96190 0.144284
$$755$$ 62.4499 2.27279
$$756$$ 0 0
$$757$$ 15.5116 0.563780 0.281890 0.959447i $$-0.409039\pi$$
0.281890 + 0.959447i $$0.409039\pi$$
$$758$$ 23.1940 0.842443
$$759$$ 0 0
$$760$$ −30.1438 −1.09343
$$761$$ −35.2085 −1.27631 −0.638154 0.769908i $$-0.720302\pi$$
−0.638154 + 0.769908i $$0.720302\pi$$
$$762$$ 0 0
$$763$$ −5.22779 −0.189259
$$764$$ −85.5129 −3.09375
$$765$$ 0 0
$$766$$ 9.25280 0.334317
$$767$$ 19.6630 0.709988
$$768$$ 0 0
$$769$$ −5.57667 −0.201100 −0.100550 0.994932i $$-0.532060\pi$$
−0.100550 + 0.994932i $$0.532060\pi$$
$$770$$ −38.5490 −1.38921
$$771$$ 0 0
$$772$$ 17.8371 0.641973
$$773$$ −35.9175 −1.29186 −0.645931 0.763396i $$-0.723531\pi$$
−0.645931 + 0.763396i $$0.723531\pi$$
$$774$$ 0 0
$$775$$ 19.1643 0.688403
$$776$$ −101.970 −3.66052
$$777$$ 0 0
$$778$$ 43.6530 1.56503
$$779$$ 3.92451 0.140610
$$780$$ 0 0
$$781$$ −6.32968 −0.226494
$$782$$ 63.7324 2.27906
$$783$$ 0 0
$$784$$ 58.8734 2.10262
$$785$$ −100.547 −3.58866
$$786$$ 0 0
$$787$$ 7.53242 0.268502 0.134251 0.990947i $$-0.457137\pi$$
0.134251 + 0.990947i $$0.457137\pi$$
$$788$$ 124.496 4.43497
$$789$$ 0 0
$$790$$ −145.666 −5.18257
$$791$$ 43.5531 1.54857
$$792$$ 0 0
$$793$$ 9.60573 0.341110
$$794$$ 73.7141 2.61602
$$795$$ 0 0
$$796$$ 91.2522 3.23435
$$797$$ 49.3837 1.74926 0.874629 0.484792i $$-0.161105\pi$$
0.874629 + 0.484792i $$0.161105\pi$$
$$798$$ 0 0
$$799$$ 12.1513 0.429883
$$800$$ 121.629 4.30025
$$801$$ 0 0
$$802$$ 94.6219 3.34122
$$803$$ −1.37759 −0.0486141
$$804$$ 0 0
$$805$$ −110.012 −3.87743
$$806$$ −6.37247 −0.224461
$$807$$ 0 0
$$808$$ 81.3549 2.86205
$$809$$ −34.4637 −1.21168 −0.605840 0.795587i $$-0.707163\pi$$
−0.605840 + 0.795587i $$0.707163\pi$$
$$810$$ 0 0
$$811$$ 20.0278 0.703272 0.351636 0.936137i $$-0.385625\pi$$
0.351636 + 0.936137i $$0.385625\pi$$
$$812$$ −17.9170 −0.628763
$$813$$ 0 0
$$814$$ −17.5429 −0.614879
$$815$$ 15.1718 0.531444
$$816$$ 0 0
$$817$$ 5.38113 0.188262
$$818$$ 96.4926 3.37379
$$819$$ 0 0
$$820$$ −77.2670 −2.69828
$$821$$ −6.45245 −0.225192 −0.112596 0.993641i $$-0.535917\pi$$
−0.112596 + 0.993641i $$0.535917\pi$$
$$822$$ 0 0
$$823$$ −43.0190 −1.49955 −0.749774 0.661694i $$-0.769838\pi$$
−0.749774 + 0.661694i $$0.769838\pi$$
$$824$$ −36.9100 −1.28582
$$825$$ 0 0
$$826$$ −125.748 −4.37533
$$827$$ 43.1557 1.50067 0.750335 0.661058i $$-0.229892\pi$$
0.750335 + 0.661058i $$0.229892\pi$$
$$828$$ 0 0
$$829$$ 13.4937 0.468656 0.234328 0.972158i $$-0.424711\pi$$
0.234328 + 0.972158i $$0.424711\pi$$
$$830$$ 57.9638 2.01195
$$831$$ 0 0
$$832$$ −11.8662 −0.411385
$$833$$ 19.9212 0.690228
$$834$$ 0 0
$$835$$ 10.7937 0.373530
$$836$$ 4.82936 0.167027
$$837$$ 0 0
$$838$$ −47.2197 −1.63118
$$839$$ 14.6851 0.506988 0.253494 0.967337i $$-0.418420\pi$$
0.253494 + 0.967337i $$0.418420\pi$$
$$840$$ 0 0
$$841$$ −27.9487 −0.963747
$$842$$ −22.4124 −0.772384
$$843$$ 0 0
$$844$$ −49.2562 −1.69547
$$845$$ 44.0858 1.51660
$$846$$ 0 0
$$847$$ 3.61829 0.124326
$$848$$ 0.994441 0.0341493
$$849$$ 0 0
$$850$$ 99.3023 3.40604
$$851$$ −50.0645 −1.71619
$$852$$ 0 0
$$853$$ −51.5775 −1.76598 −0.882990 0.469392i $$-0.844473\pi$$
−0.882990 + 0.469392i $$0.844473\pi$$
$$854$$ −61.4303 −2.10210
$$855$$ 0 0
$$856$$ −54.0755 −1.84826
$$857$$ 46.6355 1.59304 0.796520 0.604612i $$-0.206672\pi$$
0.796520 + 0.604612i $$0.206672\pi$$
$$858$$ 0 0
$$859$$ −18.6711 −0.637048 −0.318524 0.947915i $$-0.603187\pi$$
−0.318524 + 0.947915i $$0.603187\pi$$
$$860$$ −105.945 −3.61271
$$861$$ 0 0
$$862$$ −11.1888 −0.381091
$$863$$ −43.0160 −1.46428 −0.732141 0.681153i $$-0.761479\pi$$
−0.732141 + 0.681153i $$0.761479\pi$$
$$864$$ 0 0
$$865$$ 29.9462 1.01820
$$866$$ 47.5712 1.61654
$$867$$ 0 0
$$868$$ 28.8184 0.978160
$$869$$ 13.6725 0.463809
$$870$$ 0 0
$$871$$ 5.48533 0.185863
$$872$$ −10.6830 −0.361771
$$873$$ 0 0
$$874$$ 19.4898 0.659253
$$875$$ −97.6565 −3.30139
$$876$$ 0 0
$$877$$ 56.0429 1.89243 0.946217 0.323534i $$-0.104871\pi$$
0.946217 + 0.323534i $$0.104871\pi$$
$$878$$ 76.9466 2.59682
$$879$$ 0 0
$$880$$ −39.3981 −1.32811
$$881$$ −17.9947 −0.606257 −0.303128 0.952950i $$-0.598031\pi$$
−0.303128 + 0.952950i $$0.598031\pi$$
$$882$$ 0 0
$$883$$ 1.99550 0.0671540 0.0335770 0.999436i $$-0.489310\pi$$
0.0335770 + 0.999436i $$0.489310\pi$$
$$884$$ −23.3498 −0.785339
$$885$$ 0 0
$$886$$ 63.6687 2.13899
$$887$$ −7.20927 −0.242063 −0.121032 0.992649i $$-0.538620\pi$$
−0.121032 + 0.992649i $$0.538620\pi$$
$$888$$ 0 0
$$889$$ −16.9948 −0.569988
$$890$$ 129.293 4.33390
$$891$$ 0 0
$$892$$ −1.26867 −0.0424782
$$893$$ 3.71597 0.124350
$$894$$ 0 0
$$895$$ 38.9495 1.30194
$$896$$ 0.140902 0.00470720
$$897$$ 0 0
$$898$$ −101.338 −3.38168
$$899$$ −1.69102 −0.0563986
$$900$$ 0 0
$$901$$ 0.336492 0.0112102
$$902$$ 10.2559 0.341485
$$903$$ 0 0
$$904$$ 89.0006 2.96012
$$905$$ −24.5684 −0.816680
$$906$$ 0 0
$$907$$ −25.7832 −0.856116 −0.428058 0.903751i $$-0.640802\pi$$
−0.428058 + 0.903751i $$0.640802\pi$$
$$908$$ −131.965 −4.37942
$$909$$ 0 0
$$910$$ 56.9974 1.88945
$$911$$ −31.1334 −1.03149 −0.515747 0.856741i $$-0.672486\pi$$
−0.515747 + 0.856741i $$0.672486\pi$$
$$912$$ 0 0
$$913$$ −5.44061 −0.180058
$$914$$ −19.5389 −0.646291
$$915$$ 0 0
$$916$$ −26.7146 −0.882676
$$917$$ −13.5493 −0.447437
$$918$$ 0 0
$$919$$ 5.42725 0.179029 0.0895143 0.995986i $$-0.471469\pi$$
0.0895143 + 0.995986i $$0.471469\pi$$
$$920$$ −224.810 −7.41176
$$921$$ 0 0
$$922$$ −41.7560 −1.37516
$$923$$ 9.35887 0.308051
$$924$$ 0 0
$$925$$ −78.0063 −2.56483
$$926$$ −15.9469 −0.524049
$$927$$ 0 0
$$928$$ −10.7323 −0.352305
$$929$$ 21.6025 0.708756 0.354378 0.935102i $$-0.384693\pi$$
0.354378 + 0.935102i $$0.384693\pi$$
$$930$$ 0 0
$$931$$ 6.09205 0.199659
$$932$$ −132.678 −4.34603
$$933$$ 0 0
$$934$$ 69.6113 2.27775
$$935$$ −13.3313 −0.435979
$$936$$ 0 0
$$937$$ 31.6840 1.03507 0.517536 0.855661i $$-0.326849\pi$$
0.517536 + 0.855661i $$0.326849\pi$$
$$938$$ −35.0796 −1.14539
$$939$$ 0 0
$$940$$ −73.1612 −2.38626
$$941$$ 5.63987 0.183854 0.0919272 0.995766i $$-0.470697\pi$$
0.0919272 + 0.995766i $$0.470697\pi$$
$$942$$ 0 0
$$943$$ 29.2687 0.953120
$$944$$ −128.518 −4.18290
$$945$$ 0 0
$$946$$ 14.0625 0.457212
$$947$$ 51.7369 1.68122 0.840611 0.541639i $$-0.182196\pi$$
0.840611 + 0.541639i $$0.182196\pi$$
$$948$$ 0 0
$$949$$ 2.03686 0.0661194
$$950$$ 30.3674 0.985248
$$951$$ 0 0
$$952$$ 87.4850 2.83540
$$953$$ 16.2553 0.526561 0.263281 0.964719i $$-0.415196\pi$$
0.263281 + 0.964719i $$0.415196\pi$$
$$954$$ 0 0
$$955$$ 72.1874 2.33593
$$956$$ −6.77750 −0.219200
$$957$$ 0 0
$$958$$ 42.3601 1.36859
$$959$$ 57.0225 1.84135
$$960$$ 0 0
$$961$$ −28.2801 −0.912261
$$962$$ 25.9384 0.836289
$$963$$ 0 0
$$964$$ −100.304 −3.23057
$$965$$ −15.0576 −0.484721
$$966$$ 0 0
$$967$$ −54.5004 −1.75261 −0.876307 0.481754i $$-0.840000\pi$$
−0.876307 + 0.481754i $$0.840000\pi$$
$$968$$ 7.39397 0.237651
$$969$$ 0 0
$$970$$ 146.928 4.71758
$$971$$ 34.2679 1.09971 0.549855 0.835260i $$-0.314683\pi$$
0.549855 + 0.835260i $$0.314683\pi$$
$$972$$ 0 0
$$973$$ −9.14783 −0.293266
$$974$$ −12.6078 −0.403981
$$975$$ 0 0
$$976$$ −62.7834 −2.00965
$$977$$ −55.5644 −1.77766 −0.888831 0.458234i $$-0.848482\pi$$
−0.888831 + 0.458234i $$0.848482\pi$$
$$978$$ 0 0
$$979$$ −12.1357 −0.387858
$$980$$ −119.942 −3.83141
$$981$$ 0 0
$$982$$ 22.3066 0.711832
$$983$$ 41.3971 1.32036 0.660181 0.751106i $$-0.270479\pi$$
0.660181 + 0.751106i $$0.270479\pi$$
$$984$$ 0 0
$$985$$ −105.095 −3.34862
$$986$$ −8.76221 −0.279046
$$987$$ 0 0
$$988$$ −7.14054 −0.227171
$$989$$ 40.1321 1.27613
$$990$$ 0 0
$$991$$ 60.8219 1.93207 0.966036 0.258406i $$-0.0831973\pi$$
0.966036 + 0.258406i $$0.0831973\pi$$
$$992$$ 17.2623 0.548077
$$993$$ 0 0
$$994$$ −59.8515 −1.89837
$$995$$ −77.0324 −2.44209
$$996$$ 0 0
$$997$$ 26.0627 0.825414 0.412707 0.910864i $$-0.364583\pi$$
0.412707 + 0.910864i $$0.364583\pi$$
$$998$$ −35.1031 −1.11117
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1881.2.a.p.1.6 7
3.2 odd 2 209.2.a.d.1.2 7
12.11 even 2 3344.2.a.ba.1.4 7
15.14 odd 2 5225.2.a.n.1.6 7
33.32 even 2 2299.2.a.q.1.6 7
57.56 even 2 3971.2.a.i.1.6 7

By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.2 7 3.2 odd 2
1881.2.a.p.1.6 7 1.1 even 1 trivial
2299.2.a.q.1.6 7 33.32 even 2
3344.2.a.ba.1.4 7 12.11 even 2
3971.2.a.i.1.6 7 57.56 even 2
5225.2.a.n.1.6 7 15.14 odd 2